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Theorem lubfun 17448
Description: The LUB is a function. (Contributed by NM, 9-Sep-2018.)
Hypothesis
Ref Expression
lubfun.u 𝑈 = (lub‘𝐾)
Assertion
Ref Expression
lubfun Fun 𝑈

Proof of Theorem lubfun
Dummy variables 𝑥 𝑠 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funmpt 6226 . . . 4 Fun (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧))))
2 funres 6230 . . . 4 (Fun (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)))) → Fun ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧))}))
31, 2ax-mp 5 . . 3 Fun ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧))})
4 eqid 2778 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
5 eqid 2778 . . . . 5 (le‘𝐾) = (le‘𝐾)
6 lubfun.u . . . . 5 𝑈 = (lub‘𝐾)
7 biid 253 . . . . 5 ((∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)) ↔ (∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)))
8 id 22 . . . . 5 (𝐾 ∈ V → 𝐾 ∈ V)
94, 5, 6, 7, 8lubfval 17446 . . . 4 (𝐾 ∈ V → 𝑈 = ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧))}))
109funeqd 6210 . . 3 (𝐾 ∈ V → (Fun 𝑈 ↔ Fun ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦𝑠 𝑦(le‘𝐾)𝑧𝑥(le‘𝐾)𝑧))})))
113, 10mpbiri 250 . 2 (𝐾 ∈ V → Fun 𝑈)
12 fun0 6252 . . 3 Fun ∅
13 fvprc 6492 . . . . 5 𝐾 ∈ V → (lub‘𝐾) = ∅)
146, 13syl5eq 2826 . . . 4 𝐾 ∈ V → 𝑈 = ∅)
1514funeqd 6210 . . 3 𝐾 ∈ V → (Fun 𝑈 ↔ Fun ∅))
1612, 15mpbiri 250 . 2 𝐾 ∈ V → Fun 𝑈)
1711, 16pm2.61i 177 1 Fun 𝑈
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 387   = wceq 1507  wcel 2050  {cab 2758  wral 3088  ∃!wreu 3090  Vcvv 3415  c0 4178  𝒫 cpw 4422   class class class wbr 4929  cmpt 5008  cres 5409  Fun wfun 6182  cfv 6188  crio 6936  Basecbs 16339  lecple 16428  lubclub 17410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-lub 17442
This theorem is referenced by:  joinfval  17469  joinfval2  17470
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